Automorphism groups and isometries for cyclic orbit codes

Heide Gluesing-Luerssen; Hunter Lehmann

doi:10.3934/amc.2021040

Article Contents

2023, Volume 17, Issue 1: 119-138. Doi: 10.3934/amc.2021040

This issue Previous Article Upper bounds on the length function for covering codes with covering radius

$ R $

and codimension

$ tR+1 $

Next Article Construction for both self-dual codes and LCD codes

Automorphism groups and isometries for cyclic orbit codes

Heide Gluesing-Luerssen^, and
Hunter Lehmann

Department of Mathematics, University of Kentucky, Lexington, KY 40506-0027, USA

^* Corresponding author: Heide-Gluesing Luerssen
^* Corresponding author: Heide-Gluesing Luerssen

Received: February 2021

Revised: June 2021

Early access: August 2021

Published: December 2022

The first author was partially supported by the grant #422479 from the Simons Foundation

Abstract / Introduction Full Text(HTML) Related Papers Cited by

Abstract

We study orbit codes in the field extension $ \mathbb{F}_{q^n} $. First we show that the automorphism group of a cyclic orbit code is contained in the normalizer of the Singer subgroup if the orbit is generated by a subspace that is not contained in a proper subfield of $ \mathbb{F}_{q^n} $. We then generalize to orbits under the normalizer of the Singer subgroup. In that situation some exceptional cases arise and some open cases remain. Finally we characterize linear isometries between such codes.

Keywords:

Mathematics Subject Classification: Primary: 94B60.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	E. Ben-Sasson, T. Etzion, A. Gabizon and N. Raviv, Subspace polynomials and cyclic subspace codes, IEEE Trans. Inform. Theory, 62 (2016), 1157-1165. doi: 10.1109/TIT.2016.2520479.
[2]	M. Braun, T. Etzion, P. R. J. Östergård, A. Vardy and A. Wasserman, Existence of $q$-analogs of Steiner systems, Forum of Mathematics, Pi, 4 (2016). doi: 10.1017/fmp.2016.5.
[3]	B. Chen and H. Liu, Constructions of cyclic constant dimension codes, Des. Codes Cryptogr., 86 (2018), 1267-1279. doi: 10.1007/s10623-017-0394-9.
[4]	A. Cossidente and M. J. de Resmini, Remarks on Singer cylic groups and their normalizers, Des. Codes Cryptogr., 32 (2004), 97-102. doi: 10.1023/B:DESI.0000029214.50635.17.
[5]	K. Drudge, On the orbits of {S}inger groups and their subgroups, Electron. J. Combin., 9 (2002), Research Paper 15, 10 pp. doi: 10.37236/1632.
[6]	A. Elsenhans, A. Kohnert and A. Wassermann, Construction of codes for network coding, In Proc. 19th Int. Symp. Math. Theory Netw. Syst., Budapest, Hungary, 2010, 1811–1814.
[7]	T. Etzion and A. Vardy, Error-correcting codes in projective space, IEEE Trans. Inform. Theory, 57 (2011), 1165-1173. doi: 10.1109/TIT.2010.2095232.
[8]	N. Gill, On a conjecture of Degos, Cah. Topol. Géom. Différ. Catég., 57 (2016), 229-237.
[9]	H. Gluesing-Luerssen and H. Lehmann, Distance distributions of cyclic orbit codes, Des. Codes Cryptogr., 89 (2021), 447-470. doi: 10.1007/s10623-020-00823-x.
[10]	H. Gluesing-Luerssen, K. Morrison and C. Troha, Cyclic orbit codes and stabilizer subfields, Adv. Math. Commun., 9 (2015), 177-197. doi: 10.3934/amc.2015.9.177.
[11]	J. Gomez-Calderon, On the stabilizer of companion matrices, Proc. Japan Acad. Ser. A Math. Sci., 69 (1993), 140-143.
[12]	M. D. Hestenes, Singer groups, Canadian Journal of Mathematics, 22 (1970), 492-513. doi: 10.4153/CJM-1970-057-2.
[13]	B. Huppert, Endliche Gruppen, Springer, Berlin, Heidelberg, New York, 1967.
[14]	W. M. Kantor, Linear groups containing a Singer cycle, Journal of Algebra, 62 (1980), 232-234. doi: 10.1016/0021-8693(80)90214-8.
[15]	R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591. doi: 10.1109/TIT.2008.926449.
[16]	A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, In J. Calmet, W. Geiselmann, and J. M{ü}ller-Quade, editors, Mathematical Methods in Computer Science, volume 5393. Lecture Notes in Computer Science; Springer, Berlin, 2008, 31–42. doi: 10.1007/978-3-540-89994-5_4.
[17]	K. Otal and F. Özbudak, Cyclic subspace codes via subspace polynomials, Des. Codes Cryptogr., 85 (2017), 191-204. doi: 10.1007/s10623-016-0297-1.
[18]	R. M. Roth, N. Raviv and I. Tamo, Construction of Sidon spaces with applications to coding, IEEE Trans. Inform. Theory, 64 (2018), 4412-4422. doi: 10.1109/TIT.2017.2766178.
[19]	A.-L. Trautmann, Isometry and automorphisms of constant dimension codes, Adv. Math. Commun., 7 (2013), 147-160. doi: 10.3934/amc.2013.7.147.
[20]	A.-L. Trautmann, F. Manganiello, M. Braun and J. Rosenthal, Cyclic orbit codes, IEEE Trans. Inform. Theory, 59 (2013), 7386-7404. doi: 10.1109/TIT.2013.2274266.
[21]	W. Zhao and X. Tang, A characterization of cyclic subspace codes via subspace polynomials, Finite Fields Appl., 57 (2019), 1-12. doi: 10.1016/j.ffa.2019.01.002.